#### Zaph

Hello i need to determine the radius of convergense for the following series:

$$\displaystyle \sum_{n=0}^{\infty}\frac{1}{\sqrt{n!}}x^n$$

Im thinking the ratio test, but as im pretty uncertain about these types of problems, i'd like a second opinion, and a bit of help #### HallsofIvy

MHF Helper
Yes, for any "radius of convergence" problem, the best way to go is either the ratio test or the root test (and most of the time the ratio test is much simpler than the root test).

Here, $$\displaystyle a_n= \frac{1}{\sqrt{n!}}x^n$$ so $$\displaystyle a_{(n+1)}= \frac{1}{\sqrt{n+1}}x^{n+1}$$.

Then $$\displaystyle \left|\frac{a_{n}}{a_{n+1}}\right|= \frac{\sqrt{n!}}{\sqrt{(n+1)!}}|x|$$$$\displaystyle = \sqrt{\frac{n!}{(n+ 1)!}}|x|$$

What is the limit of that as n goes to infinity?
For what x is that limit less than 1?

• Zaph

#### Zaph

The limit is 1, so x < 1 will make the limit less than 1 making the radius of convergence = 1 ? #### skeeter

MHF Helper
The limit is 1, so x < 1 will make the limit less than 1 making the radius of convergence = 1 ? note that $$\displaystyle \frac{n!}{(n+1)!} = \frac{1}{n+1}$$ ... care to rethink your response?

• Zaph

#### Zaph

note that $$\displaystyle \frac{n!}{(n+1)!} = \frac{1}{n+1}$$ ... care to rethink your response?
hmm, $$\displaystyle \frac{1}{n+1}$$ diverges right?.. Im lost

#### Prove It

MHF Helper
hmm, $$\displaystyle \frac{1}{n+1}$$ diverges right?.. Im lost
$$\displaystyle \frac{1}{n + 1} \to \frac{1}{\infty} \to 0$$ as $$\displaystyle n \to \infty$$.

So what do you think the radius of convergence has to be?

• Zaph

#### Zaph

Aha! So the radius of convergence is infinity or R? since the series goes to 0.

#### Prove It

MHF Helper
Aha! So the radius of convergence is infinity or R? since the series goes to 0.
Yes, since this limit is $$\displaystyle 0$$, it is always $$\displaystyle <1$$.

So the series converges for all $$\displaystyle x$$.

• Zaph

#### Zaph

Thanx a bunch everyone, im really struggling with these types of problems, but its getting better and better 